Some algebraic and topological structures of Fourier transformable functions

Khan, Dawood; Emadifar, Homan; Ahmed, Ali; Iqbal, Saleem; Tariq, Muhammad

doi:10.22098/jfga.2024.14270.1109

Some algebraic and topological structures of Fourier transformable functions

Document Type : Original Article

Authors

¹ Department of Mathematics, University of Balochistan, Quetta, 87300, Pakistan

² Department of Mathematics, Hamedan Branch, Islamic Azad University,Hamedan, Iran

³ Department of Baic Sciences, Mehran University of Engineering and Technology, Jamshoro, Pakistan

10.22098/jfga.2024.14270.1109

Abstract

In this work, the set of all functions that are Fourier transformable with regard to their structure both algebraic and topological is taken into account. Certain topological properties of the set of Fourier transformable functions with the help of a metric are described. Also determines the proofs of the statements that the set of all Fourier transformable functions is a commutative semi-group with respect to the convolution operation as well as Abelian group with respect to the operation of addition. Metric for two functions belonging to the set of all Fourier transformable functions is defined and the proof that the Fourier transformable functions space is complete with our metric is given. The separability theorem and that the Fourier transformable functions space is disconnected are also discussed.

Keywords

References

1. K. S. Chiu and T. Li, Oscillatory and periodic solutions of differential equations with
piecewise constant generalized mixed arguments, Math. Nachr, 292(10) 2019, 2153-64.
2. L. M. Upadhyaya, On the degenerate Fourier transform, Int. J. Eng. Sci. Res, 1(6) 2018,
198-209.
3. W .R. Abd AL-Hussein and R. M. Fawzi, Solving Fractional Damped Burgers’ Equation
Approximately by Using The Sumudu Transform (ST) Method, Baghdad Sci. J., 18(1)
2021,08-03.
4. S. Aggarwal, A. R. Gupta, D. P. Singh, N. Asthana and N. Kumar, Application of
Fourier transform for solving population growth and decay problems, IJLTEMAS, 7(9)
2018, 141-5.
5. M. A Murad, Influence of MHD on Some Oscillating Motions of a Fractional Burgers
Fluid, Baghdad Sci .J., 12(1) 2015,12-22.
6. T. A. Kim and D. S. Kim, Degenerate Fourier transform and degenerate gamma function,
Russ. J. Math. Phys, 24(2)2017, 241-8.
7. Y. Kim B. M Kim , L.C Jang and J. Kwon, A note on modified degenerate gamma and
Fourier transformation, Symmetry,10 (10) 2018, 471.
8. N. Kokulan and C.H. Lai, A Fourier transform method for the image in-painting, 12th
International Symposium on Distributed Computing and Applications to Business, Engineering and Science, 2(2013), 243-246.
9. M. K. Kaabar, F. Martnez, F. F. Gmez-Aguilar, B. Ghanbari and M. Kaplan,New
approximate-analytical solutions for the nonlinear fractional Schr¨o dinger equation with
second-order spatio-temporal dispersion via double Fourier transform method, arXiv
preprint, 2(2013), 243-246.
10. S. L. Nyeo and R. R, Ansari Sparse Bayesian learning for the Fourier transform inversion
in dynamic light scattering, J. Comput. Appl. Math., 235(8) 2011, 2861-72.
11. T. Li and G. Viglialoro, Analysis and explicit solvability of degenerate tensorial problems,
Boundary Value Problems, (1)2018,1-3.
12. T. Li and G. Viglialoro, Boundedness for a nonlocal reaction chemotaxis model even in
the attraction- dominated regime, Differential and Integral Equations, 34(5/6) 2021,315-
36.
13. G. Viglialoro ,A. Gonzlez and J. Murcia, A mixed finite-element finite-difference method
to solve the equilibrium equations of a prestressed membrane having boundary cables,
International Journal of Computer Mathematics, 94(5) 2017, 933-45.